What is the fundamental difference between WebSockets and pure TCP?

It's easier to communicate via TCP sockets when you're working within an intranet boundary, since you likely have control over the machines on that network and can open ports suitable for making the TCP connections.

Over the internet, you're communicating with someone else's server on the other end. They are extremely unlikely to have any old socket open for connections. Usually they will have only a few standard ones such as port 80 for HTTP or 443 for HTTPS. So, to communicate with the server you are obliged to connect using one of those ports.

Given that these are standard ports for web servers that generally speak HTTP, you're therefore obliged to conform to the HTTP protocol, otherwise the server won't talk to you. The purpose of web sockets is to allow you to initiate a connection via HTTP, but then negotiate to use the web sockets protocol (assuming the server is capable of doing so) to allow a more "TCP socket"-like communication stream.

Web browsers operate at the Application layer, whereas TCP operates at the Transport Layer. As a web application developer, it's easier to send messages over the wire via the Application Layer instead of raw bytes at the Transport Layer.

Underlying WebSockets is TCP, it's just abstracted away for simplicity.

To make it simple, the websocket communications are done over TCP port number 80 (or 443 in the case of TLS-encrypted connections), which is of benefit for those environments which block non-web Internet connections using a firewall.

Would you like to use existed TCP port or open a new TCP port that might be blocked by firewall?

Websocket is a application layer protocol while TCP is transport layer protocol. At transport layer, we usually have TCP and UDP protocol. Any message from application layer need to go through transport layer to be transmitted to other machine. Hence, websocket and tcp have a relationship to each other and can not be comparable.